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User blog:Moooosey/Simple psi-- a fairly simple ocf
Simple psi is defined thusly (it is closely related to madore's psi, to the point that I copied the LaTeX from that page and slightly modified it): \begin{eqnarray*} C_0(\alpha) &=& \{0, 1, \Omega\}\\ C_{n+1}(\alpha) &=& \{\gamma + \delta, \psi(\eta) | \gamma, \delta, \eta \in C_n (\alpha); \eta < \alpha\} \\ C(\alpha) &=& \bigcup_{n < \omega} C_n (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*} As a demo, psi(0) = w, and psi(1) = w^2 I have a few questions about it. 1. What is psi(Ω)? 2. What is the limit of Simple psi? 3. Am I right in thinking that, for finite n, psi(n) = w^(n+1) EDIT: Thank you, @Plain'N'simple, for the quick response! Suppose I extended it thusly: (instead of doing what Buchholz did) \begin{eqnarray*} C^0_0(\alpha) &=& \{0, 1, \Omega\}\\ C^{m+1}_0(\alpha) &=& \{0, 1, \Omega,\phi(\alpha,m)\}\\ C^{m}_{n+1}(\alpha) &=& \{\gamma + \delta, \psi(\eta) | \gamma, \delta, \eta \in C^m_n (\alpha); \eta < \alpha\} \\ C^m(\alpha) &=& \bigcup_{n < \omega} C^m_n (\alpha) \\ \phi(\alpha,m) &=& \min\{\beta |\beta > \epsilon, \forall \epsilon | \epsilon \in C^m(\alpha)\} \\ C(\alpha) &=& \bigcup_{m < \omega} C^m (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*} 4. How much farther would psi get? Or would it become illdefined? If so, how would I make it work properly? 5. How would it change if I removed the m < w restriction in the penultimate line? 6. Is the current extended definition equivalent to \begin{eqnarray*} C^0_0(\alpha) &=& \{0, 1, \Omega\}\\ C^{m+1}_0(\alpha) &=& \{0, 1, \Omega,Sup (C^m(\alpha))\}\\ C^{m}_{n+1}(\alpha) &=& \{\gamma + \delta, \psi(\eta) | \gamma, \delta, \eta \in C^m_n (\alpha); \eta < \alpha\} \\ C^m(\alpha) &=& \bigcup_{n < \omega} C^m_n (\alpha) \\ C(\alpha) &=& \bigcup_{m < \omega} C^m (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*} (And for the #5 equivalent): \begin{eqnarray*} C^0_0(\alpha) &=& \{0, 1, \Omega\}\\ C^{m+1}_0(\alpha) &=& \{0, 1, \Omega,Sup (C^m(\alpha))\}\\ C^{m}_{n+1}(\alpha) &=& \{\gamma + \delta, \psi(\eta) | \gamma, \delta, \eta \in C^m_n (\alpha); \eta < \alpha\} \\ C^m(\alpha) &=& \bigcup_{n < \omega} C^m_n (\alpha) \\ C(\alpha) &=& \bigcup C^m (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*} 7. What if we add this? \begin{eqnarray*} C^0_0(\alpha) &=& \{0, 1, \Omega\}\\ C^{m+1}_0(\alpha) &=& \{0, 1, \Omega,Sup (C^m(\alpha))\}\\ C^{m}_{n+1}(\alpha) &=& \{\gamma + \delta,\gamma ^\delta, \psi(\eta) | \gamma, \delta, \eta \in C^m_n (\alpha); \eta < \alpha\} \\ C^m(\alpha) &=& \bigcup_{n < \omega} C^m_n (\alpha) \\ C(\alpha) &=& \bigcup_{m < \omega} C^m (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*} 7a. Or this? \begin{eqnarray*} C^0_0(\alpha) &=& \{0, 1, \Omega\}\\ C^{m+1}_0(\alpha) &=& \{0, 1, \Omega,Sup (C^m(\alpha))\}\\ C^{m}_{n+1}(\alpha) &=& \{\gamma + \delta,\epsilon_ {\gamma}, \psi(\eta) | \gamma, \delta, \eta \in C^m_n (\alpha); \eta < \alpha\} \\ C^m(\alpha) &=& \bigcup_{n < \omega} C^m_n (\alpha) \\ C(\alpha) &=& \bigcup_{m < \omega} C^m (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*} 7b. Or this? (Probably the best) \begin{eqnarray*} C^0_0(\alpha) &=& \{0, 1, \Omega\} \bigcup \{\alpha\}\\ C^{m+1}_0(\alpha) &=& \{0, 1, \Omega,Sup (C^m(\alpha))\}\\ C^{m}_{n+1}(\alpha) &=& \{\gamma + \delta, \psi(\eta) | \gamma, \delta, \eta \in C^m_n (\alpha); \eta < \alpha\} \\ C^m(\alpha) &=& \bigcup_{n < \omega} C^m_n (\alpha) \\ C(\alpha) &=& \bigcup_{m < \omega} C^m (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*} 7c. I'm just being weird now (Definition starts here, then goes into LaTeX) C^0_0(a) ={0, 1, W} U {a} U {b|b=psi© forall c =< a} \begin{eqnarray*}\\ C^{m+1}_0(\alpha) &=& \{0, 1, \Omega,Sup (C^m(\alpha))\}\\ C^{m}_{n+1}(\alpha) &=& \{\gamma + \delta, \psi(\eta) | \gamma, \delta, \eta \in C^m_n (\alpha); \eta < \alpha\} \\ C^m(\alpha) &=& \bigcup_{n < \omega} C^m_n (\alpha) \\ C(\alpha) &=& \bigcup_{m < \omega} C^m (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*} 8. One last thing from a comment: \begin{eqnarray*} C_0(\alpha) &=& \{0, \Omega\}\\ C_{n+1}(\alpha) &=& \{\gamma + \delta, \gamma\delta,\gamma^\delta,\phi_{\gamma}(\delta),\omega_\gamma,\psi_0(\eta) | \gamma, \delta, \eta \in C_n (\alpha); \eta < \alpha\} \\ C(\alpha) &=& \bigcup_{n < \omega} C_n (\alpha) \\ \psi_0(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \psi_1(\alpha) &=& \sup(C(\alpha)) \\ \end{eqnarray*} 8a. In response to P-bot, who said that \(\psi_1(a)\) for all a is constant (I sorta didn't care much about psi_1(a) at that time): \begin{eqnarray*} C_0(\alpha) &=& \{0, \Omega\} \bigcup_{\beta < \alpha}\{\psi_1(\beta)\}\\ C_{n+1}(\alpha) &=& \{\gamma + \delta, \gamma\delta,\gamma^\delta,\phi_{\gamma}(\delta),\omega_\gamma,\psi_0(\eta) | \gamma, \delta, \eta \in C_n (\alpha); \eta < \alpha\} \\ C(\alpha) &=& \bigcup_{n < \omega} C_n (\alpha) \\ \psi_0(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \psi_1(\alpha) &=& \sup(C(\alpha)) \\ \end{eqnarray*} EDIT: Sorry that this is in plaintext: C_0,0(α)={0,Ω} C_n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,0(η)),w_γ|γ,δ,η∈(C_n,0(α));η<α} C,0(α)=⋃(n<ω)C_n,0(α) ψ_0,0(α)=min{β∈Ω|β∉C,0(α)} ψ_1,0(α)=sup(C,0(α)) C_0,m(α)={0,Ω} C_n+1,m+1(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,m+1(η)),(ψ_1,o(γ)),(ψ_0,o(γ)),w_γ| γ,δ,η∈(C_n,m+1(α));η<α,o<(m+1)} C,m(α)=⋃(n<ω)(C_n,m(α)) ψ_0,m(α)=min{β∈Ω|β∉(C,m(α))} ψ_1,m(α)=sup((C,m(α))) Category:Blog posts